Problem 13 - College Algebra

Question

For the following exercises, find the \(x\) - and \(y\) -intercepts of the graphs of each function. $$ f(x)=2|x-1|-6 $$

Step-by-Step Solution

Answer

x-intercepts: (4, 0) and (-2, 0); y-intercept: (0, -4).

Step 1: Find the x-intercept

To find the x-intercept, set the function equal to zero and solve for x.Start with:\[2|x-1|-6 = 0\]Add 6 to both sides:\[2|x-1| = 6\]Divide both sides by 2:\[|x-1| = 3\]This gives two equations because of the absolute value:1) \(x - 1 = 3\) Solve for x: \[x = 4\]2) \(x - 1 = -3\) Solve for x: \[x = -2\]Thus, the x-intercepts are at \((4, 0)\) and \((-2, 0)\).

Step 2: Find the y-intercept

To find the y-intercept, substitute 0 for x and solve for f(x).Start with:\[f(0) = 2|0-1|-6\]Calculate the expression inside the absolute value:\[f(0) = 2|-1|-6\]Calculate the absolute value:\[f(0) = 2 imes 1 - 6\]Perform the multiplication:\[f(0) = 2 - 6\]Subtract to find:\[f(0) = -4\]Thus, the y-intercept is at \((0, -4)\).

Key Concepts

Absolute Value Functions

An absolute value function is a type of mathematical function wherein the formula involves the absolute value operator, denoted by vertical bars \(|\cdot|\). The absolute value of a number is defined as its distance from zero on the number line, disregarding its sign. For example, \(|-1| = 1\) and \(|3| = 3\). This concept plays a critical role in piecewise functions.The function given in our exercise, \(f(x) = 2|x-1|-6\), includes an absolute value, which means that the output of \(|x-1|\) is always non-negative, regardless of whether \((x-1)\) is positive or negative. In graph form, these absolute value functions often create a V-shape. The absolute value "breaks" the graph into two linear parts, one for the positive input and one for the negative input, each reflecting the sign change at the vertex or turning point.To analyze such a function, you often solve separate equations that derive from considering both the positive and negative scenarios of the expression inside the absolute value. This nature of absolute value is key to finding intercepts of the graph.

Finding X-Intercepts

The x-intercepts of a function are the points where the graph of the function crosses the x-axis. At these points, the function's value, \(f(x)\), is zero. For absolute value functions, identifying these intercepts involves solving a system that includes both the original equation and its "mirrored" form.In our exercise, we set the function equal to zero: \(2|x-1|-6 = 0\). This is an absolute value equation, meaning that we need to consider both the scenario where the contents inside the absolute value are positive and the scenario where they are negative. This results in two linear equations:

• \(x - 1 = 3\)
• \(x - 1 = -3\)

Solving these gives us the x-values where the graph intersects the x-axis. Thus, from \(x - 1 = 3\), we obtain \(x = 4\), and from \(x - 1 = -3\), we obtain \(x = -2\). Hence, the function's x-intercepts are \((4, 0)\) and \((-2, 0)\). These points indicate where the function's value is zero.

Finding Y-Intercepts

Y-intercepts are the points where the graph crosses the y-axis. At these points, the x-value is zero, and the intercept is found by evaluating the function at \(x = 0\). This value of the function gives the y-intercept, as it is simply the function's output when input \(x = 0\).For our function, \(f(x) = 2|x-1|-6\), we calculate the y-intercept by substituting \(x = 0\):

• First, calculate \(|0-1|\), which equals \(|-1| = 1\).
• Then, multiply by 2 to find \(2 \times 1 = 2\).
• Subtract 6 to solve for \(f(0)\): \(2 - 6 = -4\).

Thus, the y-intercept occurs at the point \((0, -4)\) on the graph. This means when \(x\) is zero, the output of the function, or \(f(x)\), is \(-4\), representing the point where the function intersects the y-axis. Understanding y-intercepts provides insight into how quickly or slowly a function climbs or descends as it crosses the vertical axis.